U.S. patent number 10,843,439 [Application Number 15/052,446] was granted by the patent office on 2020-11-24 for damage-resistant glass articles and method.
This patent grant is currently assigned to Corning Incorporated. The grantee listed for this patent is CORNING INCORPORATED. Invention is credited to Suresh Thakordas Gulati, Balram Suman.
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United States Patent |
10,843,439 |
Gulati , et al. |
November 24, 2020 |
Damage-resistant glass articles and method
Abstract
A strengthened glass article has opposing first and second
compressively stressed surface portions bound to a tensilely
stressed core portion, with the first surface portion having a
higher level of compressive surface stress than the second surface
portion for improved resistance to surface damage, the
compressively stressed surface portions being provided by
lamination, ion-exchange, thermal tempering, or combinations
thereof to control the stress profiles and limit the fracture
energies of the articles.
Inventors: |
Gulati; Suresh Thakordas
(Elmira, NY), Suman; Balram (Katy, TX) |
Applicant: |
Name |
City |
State |
Country |
Type |
CORNING INCORPORATED |
Corning |
NY |
US |
|
|
Assignee: |
Corning Incorporated (Corning,
NY)
|
Family
ID: |
1000005200481 |
Appl.
No.: |
15/052,446 |
Filed: |
February 24, 2016 |
Prior Publication Data
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Document
Identifier |
Publication Date |
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US 20160167343 A1 |
Jun 16, 2016 |
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Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
Issue Date |
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13083847 |
Apr 11, 2011 |
9302937 |
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61334699 |
May 14, 2010 |
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Current U.S.
Class: |
1/1 |
Current CPC
Class: |
B32B
17/06 (20130101); C03B 23/203 (20130101); C03C
23/007 (20130101); B32B 37/08 (20130101); C03C
21/00 (20130101); C03B 27/065 (20130101); B32B
37/144 (20130101); C03B 27/0413 (20130101); C03B
27/00 (20130101); B32B 2307/54 (20130101); B32B
2457/208 (20130101); Y10T 428/24992 (20150115); B32B
2307/558 (20130101); Y10T 156/10 (20150115); Y10T
428/2495 (20150115); B32B 2315/08 (20130101); B32B
37/0015 (20130101) |
Current International
Class: |
B32B
17/06 (20060101); B32B 37/08 (20060101); B32B
37/14 (20060101); C03B 23/203 (20060101); C03C
21/00 (20060101); C03B 27/00 (20060101); C03C
23/00 (20060101); C03B 27/06 (20060101); C03B
27/04 (20060101); B32B 37/00 (20060101) |
Field of
Search: |
;428/213,218 |
References Cited
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WO |
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Other References
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Materials Science and Engineering; 149; 159-165; whole document.
cited by examiner .
Eagan et al., "Bubble Formation in Glass by Reaction With Si and
Si--Ge Alloys", Journal of the American Ceramic Society, 1975, vol.
58, pp. 300-301. cited by applicant .
Giordano, et al. "Glass Transition Temperatures of Natural Hydrous
Melts: A Relationship With Shear Viscosity and Implications for the
Welding Process", Oct. 22, 2003, pp. 105-118, Journal of
Volcanology and Geothermal Research 142, Munich, Germany. cited by
applicant .
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1992, pp. 311-320, Contrib Mineral Petrol, Princeton, NY USA. cited
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Glasses and Liquids", May 25, 2004, pp. 5151-5158, Geochimica Et
Cosmochimica Acta, vol. 68, No. 24., Easton, PA USA. cited by
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Richet, et al., "Water and the Density of Silicate Glasses", Nov.
5, 1999, pp. 337-347, Contrib Mineral Petrol, Urbana, IL USA. cited
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Berkeley, CA USA. cited by applicant .
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to High-Purity Silica Glasses"; Journal of Non-Crystalline Solids
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|
Primary Examiner: Khan; Tahseen
Attorney, Agent or Firm: Hood; Michael A. Short; Svetlana
Z.
Parent Case Text
CROSS-REFERENCE TO RELATED APPLICATIONS
This application is a continuation of U.S. application Ser. No.
13/083,847 filed on Apr. 11, 2011, which claims the benefit of
priority under 35 U.S.C. .sctn. 119(e) of U.S. Provisional
Application Ser. No. 61/334,699 filed on May 14, 2010.
Claims
What is claimed is:
1. A glass article comprising: a glass core layer; first and second
glass surface layers fused to the glass core layer to form a
laminated article, each of the first and second glass surface
layers having a lower thermal expansion than the glass core layer;
and an asymmetric stress profile resulting at least partially from
subjecting the laminated article to an ion-exchange treatment at
outer surfaces of the laminated article; and a bending moment M as
defined by the expression
M=(1+.rho.)(D.sub.1+D.sub.2+D.sub.3)/.rho. of substantially zero,
wherein D.sub.1, D.sub.2 and D.sub.3 correspond, respectively, to
bending rigidities of the first glass surface layer, the glass core
layer, and the second glass surface layer, and wherein .rho.
represents a common radius of curvature of the glass article.
2. The glass article of claim 1, wherein at least one of the first
or second glass surface layers comprises a surface compression of
at least 300 MPa.
3. The glass article of claim 1, wherein the glass core layer
comprises a tensile stress not exceeding 20 MPa.
4. The glass article of claim 1, wherein the thermal expansion of
each of the first and second glass surface layers is less than or
equal to 52.times.10.sup.-7/.degree. C.
5. The glass article of claim 1, wherein the thermal expansion of
the glass core layer is greater than or equal to
74.times.10.sup.-7/.degree. C.
6. The glass article of claim 1, wherein each of the first and
second glass surface layers comprises a compressively stressed
surface portion of the glass article, and the glass core layer
comprises a tensilely stressed core portion of the glass
article.
7. The glass article of claim 1, wherein one of the first or second
glass surface layers comprises a higher surface compression than
the other of the first or second glass surface layers.
8. The glass article of claim 1, wherein the first glass surface
layer comprises a different composition than the second glass
surface layer.
9. The glass article of claim 1, wherein the first glass surface
layer comprises a different thermal expansion than the second glass
surface layer.
10. The glass article of claim 1, wherein the first glass surface
layer comprises a different thickness than the second glass surface
layer.
11. The glass article of claim 1, wherein the thermal expansion of
each of the glass core layer, the first glass surface layer, and
the second glass surface layer is the thermal expansion at a set
point of the respective layer.
12. The glass article of claim 1, wherein: at least one of the
first or second glass surface layers comprises a surface
compression of at least 300 MPa; the thermal expansion of each of
the first and second glass surface layers is less than or equal to
52.times.10.sup.-7/.degree. C.; and the thermal expansion of the
glass core layer is greater than or equal to
74.times.10.sup.-7/.degree. C.
13. The glass article of claim 1, wherein the first glass surface
layer comprises at least one of a different composition, a
different thermal expansion, or a different thickness than the
second glass surface layer.
Description
FIELD OF THE DISCLOSURE
The present disclosure relates to strengthened glass articles, and
more particularly to glass articles produced by strengthening
opposing glass surfaces of the articles by unequal amounts.
TECHNICAL BACKGROUND
The strengthening of glass articles through the introduction of
compressive stresses into the surfaces of the articles can be
accomplished by thermal tempering, ion-exchange or chemical
tempering, and the lamination of glass surface layers to glass core
layers. Thermal tempering involves rapidly cooling the surfaces of
a hot glass article to set the surface glass while allowing the
core glass to contract through slower cooling. Ion-exchange
strengthening or so-called chemical tempering typically involves
exchanging large mobile ions from the surfaces of the glass with
smaller ions in the interior of glass which can introduce
compressive stresses into the glass surfaces. In laminar
strengthening, glass surface layers or skins of relatively low
thermal expansion are fused to glass core layers of relatively high
thermal expansion so that compressive stress can develop in skins
as the laminated articles are cooled following fusion.
While each of these known methods of glass strengthening has been
employed successfully to improve the suitability of glass for a
number of existing technical applications, new applications have
imposed higher demands on the ability of glass materials to resist
surface damage in use. In some flat glass applications, for
example, the damage resistance of one of the surfaces of the glass
needs to be superior to the damage resistance of the other surface,
because one of the surfaces sees more abuse in day-to-day use than
the other surface. Touch screen displays are an example of an
application where increased surface damage resistance in the active
or exposed surface of the displays is presently required.
In most instances the known methods of glass strengthening have
been used to induce uniform compressive stresses and uniform depths
of surface compression on each of the two surfaces of the glass
article being strengthened. While resistance to surface damage can
be improved in some cases through modifications of these methods to
increase the levels of surface compression in the articles, the
results are not satisfactory for a number of applications. One
disadvantage, for example, is that increasing surface compression
can correspondingly increase core tension. High core tensions in
strengthened glass articles can undesirably increase the fracture
energy of the article in the event of breakage.
SUMMARY
In accordance with the present disclosure, strengthening methods
are employed to provide glass articles wherein one surface, termed
the primary surface, is provided with a higher level of surface
compression and/or a higher depth of surface compression layer than
the other surface. The primary surface will serve as the surface
exposed to more severe or frequent damage in service. A particular
advantage of this so-called asymmetric strengthening is that better
control over the properties of the asymmetrically strengthened
article, such as for example the level of compression in the
opposing surfaces and the level of tensile stress in the core of
the article, can be effectively controlled.
In a first aspect, therefore, the present disclosure encompasses a
glass article having opposing first and second compressively
stressed surface portions bound to a tensilely stressed core
portion, wherein the first surface portion has a higher level of
surface compression than the second surface portion. Such glass
articles are thus asymmetrically strengthened, with asymmetric
surface stresses and asymmetric stress profiles across the
thickness of the articles.
In some embodiments according to the present disclosure the surface
compression in at least the first surface portion of a disclosed
article is produced by thermal tempering; in other embodiments the
surface compression is developed through differential ion-exchange
treatment of opposing surfaces. Further embodiments include those
wherein at least the first surface portion of the strengthened
article comprises a layer of glass having a composition differing
from a composition of the core portion; such embodiments can be
provided, for example, by an ion-exchange treatment of at least the
first surface portion or by forming that surface portion through
the fusion of a layer of skin glass to the core glass of the
article.
BRIEF DESCRIPTION OF THE DRAWINGS
The articles and methods disclosed herein are further described
below with reference to the appended drawings, wherein:
FIG. 1 is a schematic orthogonal view of a flat glass plate;
FIG. 2 is an edge view of a glass plate showing a temperature
profile;
FIG. 3 is a diagram presenting stress profiles for thermally
tempered glasses;
FIG. 4 is a diagram plotting tempering stress versus cooling rate
for a glass surface;
FIG. 5 is a diagram presenting stress profiles for ion-exchanged
glasses;
FIG. 6 is a diagram plotting total stress versus surface thickness
for laminated glasses; and
FIG. 7 is a schematic elevational cross-section of a laminated
glass.
DETAILED DESCRIPTION
In general, the commercial tempering of silicate glass articles of
plate and tube configuration involves the symmetric cooling of the
articles from temperatures above the transformation range of the
glass to room temperature. The resulting stress profiles are
parabolic through the thickness of the article, with levels of
surface compression being about twice the level of central tension.
The magnitude of the central tension depends linearly on the
cooling rate R and thermal expansivity .alpha.' of the glass,
quadratically on the thickness t of the glass, and inversely on the
thermal diffusivity k of the glass, with the depth of the resulting
surface compression layers being about 21% of the thickness t. In
certain embodiments according to the present disclosure, glass
articles with opposing compressively stressed surfaces of differing
compressive stress level are provided through asymmetric thermal
tempering, such that both major surface portions of the articles
are thermally tempered, but to different degrees.
For example, the rates of cooling used to temper the opposing
surfaces can be unequal, with the first surface of the article
being cooled more rapidly than the second surface. The second
surface may be cooled more slowly than the first surface, or it may
be otherwise treated, e.g., by ion-exchange or other means, to
develop a level of compressive stress that is less than that of the
first surface. Asymmetric tempering enables improved control over
surface compressive stress levels and depths of compression layer,
thus enhancing resistance to surface damage and with only minimal
changes in stored tensile energy, i.e. fracture energy, and in
warping tendency.
The heat transfer equations applicable to thermal tempering can be
extended to asymmetric tempering both for flat plates (Cartesian
coordinates) and for cylindrical tubes (polar coordinates). The
following equations apply to the case of a glass plate where both
surfaces are thermally tempered, but where the cooling rates for
the two surfaces are unequal.
The classical differential equation for heat transfer during the
cooling of a flat glass plate is given by:
.differential..differential..times..times..differential..differential..rh-
o..times..times..times..differential..differential. ##EQU00001##
and that for a circular glass tube by:
.times..differential..differential..times..times..differential..different-
ial..rho..times..times..times..differential..differential.
##EQU00002## where T denotes temperature, t the time, K the thermal
conductivity, .rho. the density, c.sub.p the specific heat, z the
thickness coordinate for the flat plate. Assuming a constant
cooling rate R typical of a conventional commercial tempering
process, the solution of eqn. (1) subject to boundary conditions
T=T.sub.i at t=0 is given by T(z)=T.sub.i+(R/2k)z.sup.2 (3) wherein
k is the thermal diffusivity of the glass, defined as
(K/.rho.c.sub.p).
FIG. 1 of the drawings presents an isometric view of a flat glass
plate section 10 of thickness h in the x-y-z coordinate system.
Where such a plate section is cooled symmetrically on both of
surfaces S1 and S2, the temperature profile through its thickness
is parabolic and the average temperature, obtained by integrating
eqn. 3 from z=-0.5 h to z=0.5 h, is found to be:
T.sub.av=T.sub.i+[Rh.sup.2/24k] (4)
The resulting tempered glass stress distribution through the
thickness of plate section 10 is symmetric, and given at any
location z across the plate thickness by:
.sigma.(z)={E.alpha.'/(1-.nu.)}[T(z)-T.sub.av] (5) wherein .alpha.'
denotes the expansivity or coefficient of thermal expansion of the
glass in the glass transformation range, E is the Young's Modulus
of the glass, and .nu. is the Poisson's Ratio for the glass.
Equation (5) satisfies force equilibrium in the z direction. The
surface compression .sigma..sub.s and the center tension
.sigma..sub.c within the mid-plane of the glass plate (indicated by
the broken line at Z=0 in FIG. 1), follow from eqn. 5, namely
.sigma.(+/-0.5h)=.sigma..sub.s=-{E.alpha.'/(1-.nu.)}[h.sup.2R/(12k)]
(6a) .sigma.(0)=.sigma..sub.c={E.alpha.'/(1-.nu.)}[h.sup.2R/(24k)]
(6b)
Equations (6a) and (6b) confirm that the center tension is one-half
of surface compression for a symmetrically cooled plate. Further,
setting equation (5) to zero yields the depth of compression layer
.delta. as: .delta.=0.21h (7) The substantial depth of compression
secured through thermal tempering is achieved without excessive
center tension, a particularly advantageous feature where both
surface damage protection and a low fracture energy are
required.
In the case of asymmetric tempering, the cooling rate is different
on each of the two surfaces. For example, cooling may be at a rate
of R.sub.1 on a surface S1 of a plate 10 such as shown in FIG. 1 of
the drawing, and a lower rate of R.sub.2 on surface S2 of that
plate. FIG. 2 of the drawings provides a schematic illustration of
a cross-section of a glass plate such as a plate 10 upon which are
superimposed temperature profile curves that are representative of
profiles that can be developed through the use of differential
cooling rates. As shown in FIG. 2, plate 10 has a thickness h as
measured along the horizontal z axis extending from origin O, with
relative glass temperatures being reported on vertical axis T
extending from the origin and lying in the mid-plane of plate
thickness h.
The temperature profiles across glass plate 10 that result from
cooling the two plate surfaces S1 and S2 at two different rates
R.sub.1 and R.sub.2 are parabolic and asymmetric, consisting of two
different parabolas P1 and P2. Those parabolas merge at z=z.sub.o,
i.e., in a plane parallel with but offset from the mid-plane of
plate 10 by a distance z.sub.o. The higher cooling rate R.sub.1
cools a larger thickness of plate 10 (0.5 h+z.sub.o), while the
lower cooling rate R.sub.2 cools a smaller thickness of the plate
(0.5 h-z.sub.o).
Continuity conditions require that both the temperature T and the
temperature gradient at z=z.sub.o be identical, whether derived
from parabola P1 or parabola P2. Therefore,
T.sub.1(-z.sub.o)=T.sub.2(-z.sub.o)=T.sub.i, and
dT.sub.1/dz(-z.sub.o)=dT.sub.2/dz(-z.sub.o)=0 Integrating the
second equation above once, we obtain:
dT.sub.1/dz=(z+z.sub.o)R.sub.1/k for parabola P1
dT.sub.2/dz=(z+z.sub.o)R.sub.2/k for parabola P2 where k is the
thermal diffusivity of the glass as above described. Further
integration then yields:
T.sub.1(z)=T.sub.i+(z+z.sub.o).sup.2(R.sub.1/2k),-z.sub.o<z<0.5h
(8)
T.sub.2(z)=T.sub.i+(z+z.sub.0).sup.2(R.sub.2/2k),-0.5h<z<-z.sub-
.o (9) with equations 8 and 9 satisfying the required continuity
conditions. The average temperature of such an asymmetrically
cooled plate is given by
T.sub.av=T.sub.i+(R.sub.1/6kh)(0.5h+z.sub.o).sup.3+(R.sub.2/6kh)(0.5h-z.s-
ub.o).sup.3 (10) while the equations for the resulting tempering
stresses .sigma..sub.1 and .sigma..sub.2 as functions of plate
cross-sectional locations z over parabolic regions P1 and P2 are,
respectively:
.sigma..sub.1(z)={E.alpha.'/(1-.nu.)}(R.sub.1/6kh)[{3h(z+z.sub.o).sup.2-(-
0.5h+z.sub.o).sup.3-.lamda.(0.5h-z.sub.o).sup.3)}],-z.sub.o<z<0.5h
(11)
.sigma..sub.2(z)={E.alpha.'/(1-.nu.)}(R.sub.2/6kh)[{3h(z+z.sub.o).su-
p.2-(0.5h-z.sub.0).sup.3-(0.5h+z.sub.o).sup.3/.lamda.}],-0.5h<z<-z.s-
ub.o (12) wherein .lamda.=R.sub.2/R.sub.1, the ratio of the plate
surface cooling rates.
The surface compressions and stress distributions achievable
through asymmetric tempering at a selected cooling rate ratio can
be determined by measurement or calculation for any particular
glass composition selected for fabricating a glass article in
accordance with the present disclosure. FIG. 3 of the drawing
compares the calculated stress profile of a symmetrically tempered
glass plate with the calculated stress profiles for three
asymmetrically tempered plates, all four plates being of identical
3-mm thickness and soda-lime silicate glass composition. The
relevant physical properties of the glass selected for the
comparisons in FIG. 3 are reported in Table 1 below.
TABLE-US-00001 TABLE 1 Typical Physical Properties - Soda-Lime
Silica Glass Young's Modulus E (GPa) 72 Poisson's Ratio v 0.22
Coeff. Thermal Expansion .alpha.' (cm/cm/C.) 11 .times. 10.sup.-6
Thermal Diffusivity k (cm.sup.2/sec) 0.0084
The symmetrically tempered glass plate (curve 10a in FIG. 3) has a
stress profile produced by cooling both surfaces of the plate at a
cooling rate of -90.degree. C./sec. (i.e., R.sub.1=R.sub.2, or a
cooling rate ratio of 1.0). The three asymmetrically tempered
plates are tempered at cooling rate ratios R.sub.2/R.sub.1 of 0.9,
0.8 and 0.7 (Curves 10b, 10c and 10d, respectively). As the curves
in FIG. 3 reflect, the use of cooling rate ratios increasingly
below 1.0 produces increasing levels of compressive stress (higher
negative values of .sigma..sub.1 at plate surface S1), together
with increasing depths of compression layer (peak tensile stress
planes shifted toward negative values of z). As can be seen from
FIG. 3, the depth of compression layer is larger for surface S1
that experiences the higher cooling rate, and vice-versa.
From data such as shown in FIG. 3 it can be determined that the
ratio of cooling rates R.sub.2/R.sub.1 used for asymmetric
tempering should be about 0.7 or greater, to avoid the possibility
of developing tensile stress at the more slowly cooled plate
surfaces. As is known, glass fracture can easily be initiated from
surface flaws present on glass surfaces that are placed under
tension.
Yet another consideration for the case of asymmetric tempering
arises from the fact that asymmetric levels of surface compression
can cause warpage of flat glass plates, if the plates are thin and
the edges of the plates are not constrained. However, the warp
magnitudes are generally small, the warp or sagitta being readily
estimated from the equation: s=l.sup.2/8.rho. (13) wherein l
denotes the length of the plate and .rho. its radius of curvature.
The radius of curvature depends on the elastic properties of the
glass according to the equation: .rho.={E
h.sup.3/(1-.nu..sup.2)}/12M (14) wherein M, the bending moment
responsible for warp, is given by
M={E.alpha.'/(1-.nu.)}(R.sub.1/6k)[(3/64)h.sup.4+(1/4)z.sub.oh.sup.3+(3/8-
).sub.4.sup.2h.sup.2-(1/4)z.sub.0.sup.4-(R.sub.2/R.sub.1){(
3/64)h.sup.4-(1/4)z.sub.oh.sup.3+(3/8)z.sub.o.sup.2h.sup.2-(1/4)z.sub.o.s-
up.4}] (15)
In asymmetrically tempered plates wherein the plate edges are
constrained from warping, a bending moment with sign reversed from
equation (15) is introduced at plate edges. That bending moment
reduces compressive stress on one plate surface while adding a
similar amount of compressive stress on the other surface.
The advantages of asymmetric tempering and surface compression are
not limited to glass articles of plate-like configuration, but
extend to other shapes, such as cylindrical glass tubes, as well.
The integration and analysis of differential heat transfer
equations by steps analogous to those for the case of asymmetric
plate cooling as disclosed above permit calculations of the
asymmetric stress profiles resulting from the cooling of the
interiors and exteriors of glass tubes at differing cooling rates.
Embodiments of the presently disclosed glass articles that comprise
asymmetrically tempered tubing wherein high surface compression is
developed in exterior tubing surface offer particular advantages
where protection from exterior tubing damage is required.
The case of glass tubing of soda-lime silica composition with inner
radius a=2.5 cm and outer radius b=2.7 cm is illustrative. Starting
at a uniform initial glass tubing temperature of 650.degree. C.,
the outer surfaces of a series of glass tubes are cooled at a
cooling rate 90.degree. C./sec (R.sub.o=-90) while the inner
surfaces are cooled at one of a series of lower cooling rates
Cooling rate ratios R.sub.1/R.sub.o in the range of 0.1 to 1.0 are
selected for analysis.
The asymmetric stress profiles calculated from such differential
tempering treatments are reflected in FIG. 4 of the drawings. FIG.
4 presents curves plotting surface stress levels for the inner
surfaces (curve Si) and outer surfaces (curve So) of each of the
series of asymmetrically tempered tubes. The applicable cooling
rate ratios R.sub.i/R.sub.o are shown on the horizontal axis and
the resulting surface stress levels in MPa on the vertical axis,
with the more negative stress level values representing higher
surface compression in accordance with convention.
The outer surface (So) stress levels in FIG. 4 indicate that outer
surfaces of the tubes are always in compression, with rapidly
increasing compressive stress levels being achievable at lower
cooling rate ratios. The inner surface (Si) stress levels, on the
other hand, begin to decrease at similarly rapid rates, such that
inner tubing surfaces begin to experience tension at cooling ratios
below about 0.6. Thus insuring surface compression on both tubing
surfaces generally requires that cooling rate ratios of at least
0.6 be employed.
The depths .delta. of the surface compression layers developed on
the above-described tubing samples can also be calculated from the
properties of the glass and the heat transfer equations. Table 2
below sets forth inner surface (Si) and outer surface (So)
compression layer depth values .delta. for those members of the
series of asymmetrically tempered tubes of soda-lime silica
composition yielding surface compression layers on both inner and
outer tube surfaces within the given range of cooling rate ratios
R.sub.i/R.sub.o.
TABLE-US-00002 TABLE 2 Compression Layer Depths - Tempered Glass
Tubing R.sub.i/R.sub.0 .delta. - So (mm) .delta. - Si (mm) 1.0 0.43
0.40 0.9 0.45 0.38 0.8 0.52 0.26 0.7 0.56 0.16 0.6 0.62 0.01
As the Table 2 data suggest, the greater compression layer depths
.delta. on outer tubing surfaces (So), ranging from 0.43 mm to 0.63
mm, result from the higher surface cooling rates for those
surfaces, whereas the inner surface compression layer depths
.delta. range from 0.40 to as little as 0.01 mm.
Among the further embodiments of the presently disclosed glass
articles are articles wherein the surface compression in at least
one of the opposing compressively stressed surface portions
surrounding the tensilely stressed core portion is produced by an
ion-exchange treatment. Ion-exchange treatments, including
treatments wherein the surface compression in both surface portions
is developed through ion exchange treatments of both surface
portions, but to differing degrees, enable the development of
asymmetrically compressively stressed surface layers that can
exhibit stress profiles and depths of surface compression quite
different from those produced by asymmetric thermal tempering, and
advantageous for certain applications. A natural consequence of
such asymmetric ion-exchange strengthening is that the opposing
surface portions of the article have compositions differing from
each other as well as from the core portion of the article.
As was the case for asymmetric thermal tempering, equations
enabling the calculation of compressive stress levels and depths of
surface compression for glass plates or sheets can be derived from
the fundamental differential equation governing transport of mobile
ions in glasses, conventionally given by:
.differential..differential..times..times..differential..differential..di-
fferential..differential. ##EQU00003## wherein D is the ion
diffusion coefficient for the selected glass and the selected ion
(eg., Na, K, Li, etc.), C is the concentration of the exchanged
ions, z is the thickness coordinate for a flat glass plate or
sheet, and t is the ion diffusion time. The boundary conditions for
solving equations (101) are: C=C.sub.1 at z=h/2 (top surface), and
C=C.sub.2 at z=-h/2 (bottom surface) (102) with the initial
condition everywhere in the plate being: =C.sub.o at t=0 (103) and
therefore that: C.sub.o<C.sub.1 and C.sub.o<C.sub.2
Concentration equations for symmetric ion-exchange wherein the ion
concentrations at top and bottom surfaces are the same
(C.sub.1=C.sub.2) have been reported. In the symmetric case the
notation C.sub.1 can be used for both surface concentrations. A
simple known method of obtaining a solution is to use a similarity
transport. An expression for concentration in the upper half of
glass plate becomes:
.function..times..times..times..times..times.>> ##EQU00004##
where erf denotes the error function. The error function is defined
as:
.function..eta..pi..times..intg..eta..times..function..xi..times..times..-
times..times..xi. ##EQU00005## with erf(0)=0, and erf(1)=0.8427.
Similarly, an expression for concentration in the lower half of
glass plate is given by:
.function..times..times..times..times..times.>>
##EQU00006##
Since the concentration C(z) is a known function of z, the average
concentration C.sub.av can be obtained by integrating from z=h/2 to
z=-h/2:
.times..times..pi. ##EQU00007## where k=Dt/h.sup.2.
The ion-exchange stress distribution through the thickness is
symmetric and given by known equation (108) at any location z
across the plate thickness:
.sigma..function..upsilon..function..function. ##EQU00008## where B
denotes the lattice dilation constant and B[C(z)-C.sub.av]
represents the uniaxial strain induced by ion-exchange. Equation
(108) satisfies force equilibrium in the z direction.
The surface compression .sigma..sub.s and center tension
.sigma..sub.c then follow from equation (108), namely
.sigma..function..+-..sigma..upsilon..times..times..sigma..function..sigm-
a..upsilon..times..times. ##EQU00009##
As equations (109a) and (109b) suggest, the ratio of the center
tension and surface compression is the ratio of the concentration
difference between the plate mid-plane and plate surfaces relative
to the average concentration for the ion-exchanged plate. Setting
equation (109a) to zero yields the depth of compression layer
.delta., which may be obtained numerically using the known values
of C.sub.o and C.sub.1.
For the case of asymmetric ion-exchange, the exchanged ion
concentration is different at each of the two surfaces, defined as
C.sub.1 at top surface S1 and C.sub.2 at bottom surface S2. The
quantities C.sub.1 and C.sub.2 are constant. The concentration
profile, which is no longer symmetric, consists of two error
functions which merge at z=z.sub.o. The error function for C.sub.1
is valid from z=h/2 to mid-plane location z=z.sub.o and that for
C.sub.2 is valid from z=z.sub.o to z=-h/2. The location z.sub.o is
obtained by assuming that the depth over which the surface
concentration (C.sub.1 or C.sub.2) has an effect is proportional to
the concentration difference:
##EQU00010## where d.sub.1 and d.sub.2 are the depths affected by
C.sub.1 and C.sub.2, respectively. Equation (112) can be solved for
z.sub.o:
.times..times. ##EQU00011##
As expected, for C.sub.1>C.sub.2, z.sub.o is negative meaning
that C.sub.1 has effect to a greater depth (more than h/2) and vice
versa. In this case, expressions for concentration are given
by:
.function..times..times..times..times..times.>>.times..function..ti-
mes..times..times..times..times.>>.times. ##EQU00012##
The average concentration for asymmetric ion-exchange is readily
obtained by integration over the glass thickness:
.times..times..times..pi. ##EQU00013##
Similarly, the asymmetric surface stress profiles are given by
.sigma..function..upsilon..times..times..function..times..times..times..t-
imes..times.>>.times..sigma..function..upsilon..times..times..functi-
on..times..times..times..times..times.>>.times. ##EQU00014##
where the expressions for C(z) are substituted from equations
(114a) and (114b) into equation (108).
FIG. 5 of the drawings compares stress profiles for a symmetrically
ion-exchanged 1-mm thick glass plate with C.sub.1=C.sub.2=13.23 mol
% (curve 100a) against three asymmetrically ion-exchanged glass
plates. Stress levels (.sigma.) are reported in MPa on the vertical
graph axis and plate cross-sectional location z on the horizontal
axis. The asymmetrically ion exchanged plates are of the same
geometry and base glass composition and have the same first surface
(S1) exchanged ion concentration as the symmetrically ion-exchanged
plate, but with varying lower exchanged-ion concentrations on the
second plate surface (S2), i.e., with C.sub.1=13.23 mol % and
C.sub.2=10.23, 5.23, or 2.7 mol % (curves 100b, 100c and 100d,
respectively). The exchanged-ion concentration in the base glass
composition (C.sub.0) is 2.55 mol %. The physical properties of the
base glass and the diffusion time (t) employed to reach the C.sub.1
exchanged-ion concentration are reported in Table 3 below.
TABLE-US-00003 TABLE 3 Physical properties and ion-exchange
parameters Young's Modulus E (MPa) 72900 Poisson's Ratio .upsilon.
0.211 B (m/m/mol %) 11 .times. 10.sup.-6 Diffusivity D
(m.sup.2/sec) .sup. 1.1 .times. 10.sup.-14 t, sec 28800 Thickness,
mm 1
As the stress profile curves in FIG. 5 reflect, decreasing
exchanged-ion concentrations at S2 result in decreasing S2
compressive stress levels .sigma. and decreasing depths .delta. of
compression layer at that surface, although the difference in
compression layer depths .delta..sub.t and .delta..sub.b as between
the top and bottom surfaces is not large. For example, where the
ratio of exchanged-ion concentration (C.sub.1) at top surface S1 to
the exchanged-ion concentration (C.sub.2) at bottom surface S2 is
about 0.6, the top surface compression level is approximately 800
MPa and the bottom surface compression level is approximately 400
MPa. At those compression levels the depths of the compression
layers are approximately 0.054 mm and 0.047 mm, respectively. Thus
both surfaces incorporate relatively deep compression layers, a
characteristic that is particularly advantageous when protection
against bottom surface as well as top surface damage is
required.
The analyses of asymmetric ion-exchange strengthening set forth
above do not account for plate bending contributions since the
plates being characterized are not edge-constrained. As was the
case for asymmetric thermal tempering, edge constraints will modify
the stress distribution profile of a glass plate comprising
ion-exchanged surface compression layers, generally by increasing
surface compression on the plate surface having the highest
unconstrained compressive stress level and decreasing surface
compression on the opposite plate surface. However, for the case of
asymmetric ion-exchange strengthening, analyses based on the
elastic properties of glass indicate that, even for modest plate
thicknesses, warp is sufficiently small that it is not problematic
for most applications.
Still further embodiments of the presently disclosed glass articles
are articles provided by laminating the surface portions to a glass
core differing in composition and physical properties from the
surface portions, but wherein the laminated surface portions
exhibit differing levels of compressive stress. For example,
asymmetric lamination embodiments can provide glass articles
wherein the opposing compressively stressed surface portions joined
to the tensilely stressed core are of differing thicknesses, or
differing compositions and thermal expansivities, thus providing
compressive stress in a first surface portion that differs from the
compressive stress in a second surface portion. The characteristics
of such articles compare favorably with those of articles
comprising asymmetric cross-sectional stress profiles provided by
thermal or chemical ion-exchange tempering or strengthening, and
offer further advantages for certain applications.
For purposes of the following analysis reference is made to FIG. 7
of the drawings schematically showing an elevational diagram of a
laminated article, not in true proportion or to scale. The article
comprises a first surface layer or skin L1, a core layer L2, and a
second surface layer or skin L3, those components being of
thicknesses h.sub.1, h.sub.2 and h.sub.3 respectively.
Embodiments of laminated glass articles in accordance with the
present description include those wherein both of the surface or
skin portions or layers of the laminate are composed of the same
glass, but wherein the thickness of one skin layer differs from
that of the other skin layer. In those embodiments each of the
three glass layers (first surface layer L1, core layer L2 and
second surface layer L3) will experience bending moments M.sub.1,
M.sub.2, and M.sub.3 respectively, due to the differing direct
compressive forces N.sub.1 and N.sub.3 and tensile force N.sub.2
that develop in the skins and core, respectively, during the
cooling and contraction of the layers of the article following
high-temperature lamination. Again the unconstrained laminates will
exhibit some cylindrical warping due to their asymmetric stress
profiles, but the amounts of warp are generally small.
As the following analysis will show, the expressions for forces
N.sub.1, N.sub.2, and N.sub.3, and bending moments M.sub.1, M.sub.2
and M.sub.3 developed below satisfy the applicable force
equilibrium and Moment/warp relationships, namely
N.sub.1+N.sub.2+N.sub.3=0 (201)
M.sub.1+M.sub.2+M.sub.3=M=(1+.rho.)(D.sub.1+D.sub.2+D.sub.3)/.rho.
(202) where the D values denote the bending rigidities of the
layers and .rho. represents the common radius of curvature of the
structure where warping due to asymmetric surface compression
levels occurs.
The pertinent equations for direct stress (a.sub.d) and bending
stress (a.sub.b) are summarized as follows:
.sigma..sub.d1=N.sub.1/h.sub.1 (203)
G.sigma..sub.d2=N.sub.2/h.sub.2 (204)
.sigma..sub.d3=N.sub.3/h.sub.3 (205)
.sigma..sub.b1=6M.sub.1/h.sub.1.sup.2 (206)
.sigma..sub.b2=6M.sub.2/h.sub.2.sup.2 (207)
.sigma..sub.b3=6M.sub.3/h.sub.3.sup.2 (208) wherein h.sub.1,
h.sub.2 and h.sub.3 are the thicknesses of the first surface layer
L1, core layer L2, and second surface layer L3, respectively, and
wherein
N.sub.1=[{(a.sub.3+b.sub.2+b.sub.3)}(.alpha..sub.c-.alpha..sub.s)(T.sub.s-
et-25)]/.DELTA. (209)
N.sub.3=[{(a.sub.1+b.sub.1+b.sub.2)}(.alpha..sub.c-.alpha..sub.s)(T.sub.s-
et-25)]}/.DELTA. (210) N.sub.2=(N.sub.1+N.sub.3)
.DELTA.=(a.sub.2-b.sub.2).sup.2-(a.sub.1+a.sub.2+b.sub.1)(a.sub.2+a.sub.3-
+b.sub.3) (211) M.sub.1={D.sub.1/(D.sub.1+D.sub.2+D.sub.3)}M (212)
M.sub.2={D.sub.2/(D.sub.1+D.sub.2+D.sub.3)}M (213)
M.sub.3={D.sub.3/(D.sub.1+D.sub.2+D.sub.3)}M (214)
M=0.5[N.sub.1(h.sub.1+h.sub.2)-N.sub.3(h.sub.2+h.sub.3)] (215)
with: .alpha..sub.c and .alpha..sub.s being the thermal expansion
coefficients of the core and skin layers at the thermal set points
of the glasses making up those layers, and with a.sub.1, a.sub.2,
a.sub.3, b.sub.1, b.sub.2 and b.sub.3 being constants related to
the skin and core layer thicknesses h.sub.1, h.sub.2 and h.sub.3,
Young's Modulus E, and Poisson's Ratios .nu. of the core and skin
glasses as: a.sub.1=(1-.nu.)/(E.sub.1h.sub.1) (216)
a.sub.2=(1-.nu.)/(E.sub.2h.sub.2) (217)
a.sub.3=(1-.nu.))/(E.sub.3h.sub.3) (218)
b.sub.1=(h.sub.1+h.sub.2).sup.2/[4(1+.nu.(D.sub.1+D.sub.2+D.sub.3)]
(219)
b.sub.2=(h.sub.1+h.sub.2)(h.sub.2+h.sub.3)/[4(1+.nu.))(D.sub.1+D.su-
b.2+D.sub.3)] (220) b3=(h2+h3)2/[4(1+.nu.)(D1+D2+D3)] (221)
For any particular laminated glass article of the kind herein
described, the total stress in each of the three layers is simply
the sum of the direct and bending stresses in that layer. The total
stresses for layers L1, L2 and L3 are given respectively by:
.sigma..sub.1=(N.sub.1/h.sub.1)+(6M.sub.1/h.sub.1.sup.2) (222)
.sigma..sub.2=(N.sub.2/h.sub.2)+(6M.sub.2/h.sub.2.sup.2) (223)
.sigma..sub.3=(N.sub.3/h.sub.3)+(6M.sub.3/h.sub.3.sup.2) (224)
For the asymmetrically stressed surface layers of the strengthened
articles of the present disclosure, the direct stresses
(N.sub.1/h.sub.1) and (N.sub.3/h.sub.3) are compressive or negative
skin stresses and the direct stress (N.sub.2/h.sub.2) is a tensile
or positive core stress. Similarly, where the first surface layer
has a thickness h.sub.1 that is greater than h.sub.3, bending
moments M.sub.1 and M.sub.3 have a sign such that
(6M.sub.1/h.sub.1).sup.2 and (6M.sub.3/h.sub.3).sup.2 are
compressive or negative stresses on one of the surfaces of first
and third layers. The tensile component (6M.sub.2/h.sub.2).sup.2
will occur either at top of core layer L2 or the bottom of that
core layer depending on whether layer L3 is thicker than layer L1
or vice-versa. In short, the tensile stress in the core layer is
the sum of direct and bending stresses in that layer.
Advantageously, either layer L1 or layer L3 will experience maximum
compressive stress on their exposed outer surfaces where maximum
damage resistance is desired. As noted above, the depths of surface
compression are simply the respective thicknesses h.sub.1 and
h.sub.3 of the compressively stressed surface layers.
As in other asymmetrically stressed glass articles provided in
accordance with the present disclosure, unequal skin thicknesses
h.sub.1 and h.sub.3 produce an asymmetric stress distribution that
tends to warp the article into a cylindrical shape of radius p.
That radius is given by
.rho.=D.sub.1(1+.nu.)/M.sub.1=D.sub.2(1+.nu.))/M.sub.2=D.sub.3(1+.nu.))/M-
.sub.3 (225) with the sagitta or maximum warp being given by
.delta.=L.sup.2/(8.rho.) (226) wherein L denotes the longer
dimension of 3-layer laminate.
The use of the foregoing analyses to calculate the asymmetric
stress distributions that can be generated in laminated glass
articles is illustrated below for the case of a three-layer article
wherein the compressively stressed surface layers are of the same
composition but different thicknesses. For the purpose of analysis
the total thickness of the laminated article and the thickness
h.sub.2 of the glass core are kept constant while the skin
thicknesses h.sub.1 and h.sub.3 are varied to determine the effects
of the asymmetric geometries on surface compression levels and the
levels of tensile stress in the core portion. As a specific
example, a laminate of 2.677 mm total thickness (h1+h2+h3) and 2.54
mm core thickness is analyzed.
The physical properties of skin and core glasses for the laminated
article are set forth in Table 4 below. Included in Table 4 for
each of the core and skin glasses are the elastic or Young's
Modulus, in GPa, the Poisson's Ratio, the Glass Set Point or
temperature at which the glass solidifies on cooling, in .degree.
C., and the thermal expansion of each glass at the Set Point.
TABLE-US-00004 TABLE 4 Properties of Core and Skin Glasses Young's
Glass Thermal Expanion Glass Modulus E Poisson's Set Point .alpha.
at Set point Component (GPa) Ratio .upsilon. (.degree. C.)
(10.sup.-7/C) Core 75.0 0.22 562 74.5 Skin 85.5 0.22 675 52.0
The glass articles evaluated included glass laminates wherein the
ratio of L1 surface layer thickness to L3 surface layer thickness
ranged from 1 to 2. Results of these evaluations for a number of
laminated glass articles incorporating asymmetric surface layers
are set forth in FIG. 6 of the drawings. FIG. 6 plots the total
stresses calculated on outer surfaces of layers L.sub.1 and L.sub.3
as well as in layer2 for each of the three layers for each of ten
laminated articles having surface layer thickness ratios
h.sub.3/h.sub.1 in the above range.
The total stress values plotted in FIG. 6 are combinations of the
direct stresses and bending stresses in each layer in accordance
with the above analyses. The observed trends in L1 and L3 total
stress levels are due mainly to the same trends in the direct
stress levels calculated for those layers. The bending stresses in
the surface layers are calculated to be small due to the low
bending rigidities (D.sub.1 and D.sub.3) of the surface layers.
Particularly advantageous features of the asymmetrically stressed
laminates of these embodiments of the present disclosure are the
relatively modest center tensions maintained in the cores
notwithstanding the relatively high compressive stresses developed
in the surface layers. This combination of features provides
excellent surface damage resistance without undesirable increases
in tensilely stressed fracture energy of layer L.sub.2. It would
permit the strengthened glass articles to be scored and separated
safely with little risk of long-term fatigue of exposed edges of
the separated sections. In addition, the maximum warp values
calculated for the most highly stressed laminates, i.e., less than
0.1 mm over a 10-cm laminate span, would not be of concern for most
applications.
As the foregoing analyses suggest, asymmetrically strengthened
glass sheet products, including for example glass display panels or
display cover sheets, or even glass panes employed as glazing
elements in other devices or structures, are of particular interest
where the intended application involves exposure to repeated
physical contact or a high risk of impact damage. For these and
other applications, however, it is generally important that the
sheet surface exhibiting the lower level of compressive surface
stress not be subjected to tensional stress, physical tension, and
that the level of tension in the core of the sheet not be so great
as to develop unacceptably high fracture energy in the strengthened
glass product.
In light of these considerations specific embodiments of the
presently disclosed articles particularly include glass sheets
having opposed compressively stressed first and second surfaces
joined to a tensiley stressed interior core, wherein the first
surface has a higher level of surface compression than the second
surface, and wherein the core has a tensile stress level not
exceeding about 20 MPa. Embodiments of such sheets wherein the
level of compressive stress in the compressively stressed second
surface is at least 300 MPa, both for the unconstrained sheet and
for the sheet when edge-constrained to remove sheet warping due to
asymmetric surface stresses.
Methods for making glass articles such as strengthened glass sheets
or vessels in accordance with the foregoing disclosure are carried
out utilizing apparatus and materials employed in the prior art for
tempering, ion-exchanging and/or laminating heated or softened
glasses. For the manufacture of a strengthened glass sheet having
opposing first and second compressively stressed surfaces bound to
a tensilely stressed core with the first surface having a higher
level of surface compression than the second surface, the glass
sheet is cooled from a temperature above the glass set point in a
manner such that the first and second surfaces are cooled at
different cooling rates. In particular embodiments, for example,
the first surface of the sheet is cooled at a cooling rate R.sub.1
and the second surface is cooled at a lower cooling rate R.sub.2,
and the ratio R.sub.2/R.sub.1 is selected to be at least at least
0.7.
For the manufacture of strengthened glass sheet having an
asymmetric stress profile via ion-exchange, a step of subjecting
the first and second surfaces of the sheet to different
ion-exchange strengthening treatments is used. Thus the first
surface is subjected to a first ion-exchange treatment and the
second surface is sequentially or concurrently subjected to a
second ion-exchange treatment differing from the first treatment.
In particular embodiments, the first ion-exchange treatment
develops a concentration C.sub.1 of exchanged ions in the first
surface of the sheet and the second ion-exchange treatment develops
a concentration C.sub.2 of exchanged ions in the second surface of
the sheet, with C.sub.1 differing from C.sub.2 to a degree
effective to develop a selected asymmetric stress profile
characterized by differing levels of surface stress in the
glass.
The manufacture of strengthened glass sheets or other articles
having asymmetric surface compression in opposing surfaces does not
require the use of only one strengthening strategy to achieve the
desired stress profiles. Rather, these strengthening methods can be
advantageously combined to develop asymmetric profiles not
achievable using any one strategy alone.
As one example, a strengthened glass sheet having opposing first
and second compressively stressed surfaces of differing stress
level can be made by the steps of first cooling the surfaces from a
temperature above the glass set point to develop first and second
levels of compressive stress in those surfaces, and then subjecting
the first and second surfaces to ion-exchange strengthening to
modify at least one of the first and second compressive stresses.
At least one of these steps will be carried out in a manner that
develops a higher level of compressive stress in the first surface
than in the second surface.
A further example of a combination method for making a strengthened
sheet or other article comprises a first step of laminating first
and second glass skin layers to a glass core layer to form a
laminated glass sheet having first and second levels of compressive
stress in opposing first and second surfaces of the sheet; and a
second step of subjecting the first and second surfaces to
ion-exchange strengthening to modify at least one of the first and
second compressive stresses. Again, at least one of the two steps
is conducted to develop a higher level of compressive stress in the
first surface than in the second surface.
Yet another embodiment of a combination method for manufacturing a
strengthened glass sheet having a first surface having a higher
level of surface compression than a second surface comprises the
steps of laminating first and second glass skin layers to a glass
core layer to form a laminated glass sheet having first and second
levels of compressive stress in opposing first and second surfaces
of the sheet, and then cooling the first and second surfaces from a
temperature above the set points of the surfaces to modify at least
one of the first and second compressive stresses. Either the
lamination step, or the cooling step, or both, will be carried out
in a way that develops a higher level of compressive stress in the
first surface than in the second surface.
Further embodiments of the disclosed asymmetrically stressed glass
articles include articles essentially free of stress-induced
warping that still retain an asymmetric stress profile. As above
equation (226) suggests, sagitta or maximum warp .delta. vanishes
when the radius of curvature .rho. of a surface-stressed article
approaches infinity, i.e., when bending moment M from equation
(202) above approaches zero. Of course a trivial solution meeting
those conditions is that of a symmetrical laminate with opposing
surface portions (e.g., skin glass layers) of identical thickness
and composition. In that case the thermal expansions and elastic
moduli of the layers are necessarily the same, such that no
asymmetric stress profile is present.
In accordance with the present disclosure, however, articles
substantially free of stress-induced warping, but still offering
asymmetric stress profiles, are provided through the use of
opposing first and second compressively stressed surface portions
that differ in both composition and thickness. The compositions and
thicknesses selected are those effective to generate differing
surface compression levels, but that do not introduce changes in
curvature or flatness in the stressed articles. Examples include
surface-stressed articles such as laminated articles that have a
bending moment M as defined by the expression
M=(1+.rho.)(D.sub.1+D.sub.2+D.sub.3)/.rho. of substantially zero,
wherein D.sub.1, D.sub.2 and D.sub.3 correspond, respectively, to
the bending rigidities of the first surface portion, a glass core,
and a second surface portion, and wherein .rho. represents the
common radius of curvature of the stressed article arising from
warping due to asymmetric surface compression levels.
To arrive at a numerical expression defining the thicknesses,
thermal expansions, and elastic moduli suitable for providing such
articles, the expressions for N.sub.1 and N.sub.3 from equations
(209) and (210) above are substituted into equation (215) above to
yield:
(h.sub.1+h.sub.2)(a.sub.3+b.sub.2+b.sub.3)(.alpha..sub.c-.alpha..sub.s1)=-
(h.sub.2+h.sub.3)(a.sub.1+b.sub.1+b.sub.2)(.alpha..sub.c-.alpha..sub.s2)
(227)
Then, setting .nu..sub.1=.nu..sub.2=.nu..sub.3=.nu., which is valid
for most silicate glasses, it can be shown from equations (216) to
(221) that:
a.sub.3+b.sub.2+b.sub.3=(1-.nu.)[E.sub.1h.sub.1.sup.3+E.sub.2h.sub.-
2.sup.3+E.sub.3h.sub.3.sup.3+3E.sub.3h.sub.3(h.sub.1h.sub.2+2h.sub.2.sup.2-
+3h.sub.2h.sub.3+h.sub.1h.sub.3+h.sub.3.sup.2)]/.DELTA..sub.1 (228)
and
a.sub.1+b.sub.1+b.sub.2=(1-.nu.)[E.sub.1h.sub.1.sup.3=E.sub.2h.sub.2.sup.-
3+E.sub.3h.sub.3.sup.3+3E.sub.3h.sub.3(h.sub.1h.sub.2+2h.sub.2.sup.2+3h.su-
b.2h.sub.3+h.sub.1h.sub.3+h.sub.3.sup.2)]/.DELTA..sub.2 (229)
wherein
.DELTA..sub.1=E.sub.3h.sub.3(E.sub.1h.sub.1.sup.3+E.sub.2h.sub.2.sup.3+E.-
sub.3h.sub.3.sup.3) and
.DELTA..sub.2=E.sub.1h.sub.1(E.sub.1h.sub.1.sup.3E.sub.2h.sub.2.sup.3E.su-
b.3h.sub.3.sup.3)
Substituting the above expressions in equation (227) above, and
simplifying, yields the following relationship between the
expansion mismatches, elastic moduli, and stressed surface portion
(e.g., skin glass) thicknesses for a family of asymmetrically
stressed glass articles substantially free of surface warping:
(.alpha..sub.c-.alpha..sub.s1)/(.alpha..sub.c-.alpha..sub.s2)={(h.sub.2+h-
.sub.3)/(h.sub.2+h.sub.1)}[h.sub.1.sup.2+(E.sub.2/E.sub.1)(h.sub.2.sup.3/h-
.sub.1)+(E.sub.3/E.sub.1)(h.sub.3.sup.3/h.sub.1)+3(h.sub.1.sup.2+3h.sub.1h-
.sub.2+h.sub.1h.sub.3+2h.sub.2.sup.2+h.sub.2h.sub.3)]/[h.sub.3.sup.2+(E.su-
b.2/E.sub.3)(h.sub.2.sup.3/h.sub.3)+(E.sub.1/E.sub.3)(h.sub.1.sup.3/h.sub.-
3)+3(h.sub.3.sup.2+3h.sub.3h.sub.2+h.sub.1h.sub.3+2h.sub.2.sup.2+h.sub.2h.-
sub.1)] (230)
While the above equation clearly encompasses the case where
h.sub.1=h.sub.3 and E.sub.1=E.sub.3, (i.e., the case of identical
skin properties and skin thicknesses), a large group of
asymmetrically stressed laminates or other articles wherein the
thicknesses and elastic modulii of the opposing skin layers or
surface portions are dissimilar is also defined. The latter group
can be generally characterized as articles that include three
different glass compositions, selected to provide a high
compression first surface portion or skin, a core, and a lower
compression second surface portion or skin, such that warping of
the article is mitigated even though at least one highly stressed,
damage resistant surface layer is provided.
Of course, the foregoing descriptions and specific embodiments of
the disclosed articles and methods are presented for purposes of
illustration only, it being apparent from those descriptions that a
wide variety of adaptations and modifications of those particularly
disclosed embodiments may be adopted to meet the requirements of a
variety of applications within the scope of the appended
claims.
* * * * *